Basic Description

Anne Burns' Mathscape images are natural looking landscapes whose components are created using math. The plants are generated through recursive algorithms similar to those used to make the plant-like fractal Blue Fern, and through string rewriting. The mountains and clouds are created using height fields (see below) and trigonometry.

A More Mathematical Explanation

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Plants

Some of the algorithms Burns uses to create plants are similar to the fract [...]

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Plants

Some of the algorithms Burns uses to create plants are similar to the fractal algorithms used to generate the Blue Fern. Unlike the Blue Fern, however, Burns' plants aren't perfect fractals that are exactly self-similar into infinity. Instead, some aspects of growth are randomized, so that the final image looks more natural.

Image 2 shows some basic plant formations similar to those that Burns used in her replacements. Each offshoot in this figure is replaced by a smaller version of the original image, which is again replaced by a smaller version, a process that repeats until the picture is quite complicated. The pictures can be made to look more natural by using random numbers to vary the length of the branches or the angle at which they connect.

The plant images can also be made to look more natural by using the deReffye method. In this method, at each iteration of the growth algorithm a branch can either continue to grow, die, or do nothing until the next iteration. Each option is assigned a certain probability, and the probabilities change over time. At later stages the branches are more likely to die, which means that plant eventually stops growing.

The trees in Image 3 were created from the same program using the deReffye method. Because they're generated by the same algorithm, they look similar, but because of the possibility for variation built into the program, their shapes are not identical.

Burns also uses string re-writing to create images of plants. A plant created using this method can be seen in Image 4. In string-rewriting, also called an L-system, a simple plant image composed of sticks, leaves, and flowers is represented by a string of letters and punctuation marks.

For example, Burns describes a system where the letter I represents a stem, the letter L represents a leaf, and the letter F represents a flower. Branches to the left are enclosed with by brackets, and branches to the right are enclosed by parentheses. Some simple plant images and the strings that describe them are shown in Image 5:

At each stage, a set of substitution rules replaces each character with a string of other characters. The end result string is then translated back into a more complicated plant image. For example, there might be a rule that replaces the string L with the string I(L)F. Applying this rule once to the flower images in Image 5 yields:

Image 6. The plants from Image 5 after the replacement rule L → I(L)F has been applied once.

Notice that the second plant didn't change, because it's string didn't contain an L. Also, it is important to note that this particular replacement rule could be carried out infinitely, since I(L)F, the replacement string, contains L, the original string. Several replacement rules can also be used in combination.

Clouds & Mountains

Burns explains how a simple image that looks like the silhouette of a mountain can be generated by a midpoint replacement rule that uses random displacement. The progression of such a rule is shown below:

In the first stage, we have endpoints with two different heights, and a dotted line connecting them. In the second stage, we find the midpoint of the dotted line, and change its height by a random amount (in either direction). We now have two intervals, and in the next step we find and change the midpoints of both of them. As this process is repeated, the image begins to look more and more like a mountain.

This method only generates a 2-D image, or the mountain's silhouette, but the 3-D clouds and the mountains in Burns' images have essentially the same math behind them. The idea is to create something that Burns calls a height field, which can either be assigned colors to look like clouds or projected as a three-dimensional image into two-dimensional space to create mountains.

The first step is to create a grid, and assign values to the four corner points. Next, the center point is assigned to be the average of the four corners plus or minus a random number. In each stage, the grid is further subdivided, and the points at each location are assigned values based on nearby points, the same way the center point was. Over time, the random numbers added to the heights are scaled down. The rate at which these numbers scale down affects the shape of the height field, and manipulating it can produce jagged mountains and wispy clouds or rounded mountains and soft clouds.

The first couple stages of assigning values to the height field are shown below:

When making clouds, the values assigned to each point are interpreted along a color spectrum, determining how light or dark that area of the sky is. To make mountains, the values are interpreted as heights. With mountains, there is also the additional step of changing the perspective. The values assigned to the height field give us the z-coordinates of each point, but in the image above, the z-axis is coming out of the screen, giving a bird's eye view of the mountains. To use the height field as a mountain in an image, trigonometry and calculus are used to change the perspective of the mountain, so that the z-axis is going up and down.

Why It's Interesting

Burns' images reveal the fractal geometry behind many natural phenomena. They are excellent examples of the close connections between math, art, and nature. The landscapes she's created are beautiful and natural, and it can be surprising to learn that they were created through mathematics and computer programming.

Teaching Materials

About the Creator of this Image

Anne M. Burns is a professor at Long Island University's C.W. Post campus. She received her Ph.D. in Mathematics from SUNY Stony Brook in 1976. Her research interests include discrete dynamical systems, scientific visualization, and using mathematics and computer graphics to describe nature.